1.Write the equation of the conic section with the given properties:
A hyperbola with vertices(0,6)(0,-6)and asymptotes y=3/4x and y=-3/4x.
2.Write the equation of the conic section with the given properties:
An ellipse with vertices(0,-5)(0,5)and a minor axis of length 8.
Im having major trouble i got x^2=4r2 for 1 and x^2=49 for 2
To find the equations of the given conic sections, you need to understand the standard forms for hyperbolas and ellipses.
1. For the hyperbola:
A hyperbola with vertices (0,6) and (0,-6) has a transverse axis along the y-axis, and the distance from the center to each vertex is called "a". Since the vertices are on the y-axis, the center of the hyperbola is at the origin (0,0).
The equation of a hyperbola with a vertical transverse axis and centered at the origin is given by:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1.
Here, (h,k) represents the center point. In this case, h = 0 and k = 0.
To determine the values of a and b, note that the distance between the center and each vertex is a. In this case, the distance between the center and each vertex is 6. Therefore, a = 6.
Now, we need to find the value of b. The asymptotes of the hyperbola are y = 3/4x and y = -3/4x. The slopes of the asymptotes are given by the ratio of b to a, i.e., b/a.
The slope of the asymptotes is 3/4, so b/a = 3/4. Since a = 6, we can solve for b:
b/a = 3/4
b/6 = 3/4
Cross-multiplying, we get:
4b = 18
b = 18/4
b = 4.5
Now, we can substitute the values of a, b, and the center into the equation to get the final equation:
(y - 0)^2 / 6^2 - (x - 0)^2 / 4.5^2 = 1
Simplifying,
y^2 / 36 - x^2 / 20.25 = 1
This is the equation of the hyperbola with the given properties.
2. For the ellipse:
An ellipse with vertices (0,-5) and (0,5) has a transverse axis along the y-axis, and the distance from the center to each vertex is called "a". Since the vertices are on the y-axis, the center of the ellipse is at the origin (0,0).
The equation of an ellipse with a vertical transverse axis and centered at the origin is given by:
(y - k)^2 / a^2 + (x - h)^2 / b^2 = 1.
Again, (h,k) represents the center point. In this case, h = 0 and k = 0.
To determine the value of a, note that the distance between the center and each vertex is a. In this case, the distance between the center and each vertex is 5. Therefore, a = 5.
The length of the minor axis is 8, which is twice the value of b. Hence, b = 8/2 = 4.
Now, substitute the values of a, b, and the center into the equation to get the final equation:
(y - 0)^2 / 5^2 + (x - 0)^2 / 4^2 = 1
Simplifying,
y^2 / 25 + x^2 / 16 = 1
This is the equation of the ellipse with the given properties.